When I imagine putting two apples next to two apples, I can predict what will actually happen when I put two earplugs next to two earplugs, and indeed, my mind can store the result in a generalized fashion which makes predictions in many specific instances. If you do not call this useful abstract belief “2 + 2 = 4”, I should like to know what you call it. If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.
No, the real world does not work via Peano arithmetic. Your experiments with apples and earplugs are simply applications of conservation of mass and immutability of inanimate objects, and other such principles. Before you learned such things, you were thrilled with the game of peek-a-boo—of how someone could cease to exist, and then appear out of nowhere.
Consider this experiment: Take 2 apples, cut them in half. Take 2 more apples, cut them in half. Put all together. How many apples do you have? The answer is not “4 apples”, the answer is “8 half-apples”. Furthermore, each individual apple remains the same apple as before (minus the effects of time), so that any differences in size, shape, coloration, bruising, etc would remain the same. Apples aren’t numbers, and can’t be substituted for each other.
The world abounds with examples where Peano arithmetic does not apply. Consider adding two speeds together—they do not add via Peano arithmetic, such that there exists a speed X, such that 2X + 2X = 3X. If we’re using naive multiplication, that speed is where X=c*sqrt(7/36). None of this changes my beliefs about Peano arithmetic—it is necessarily true given its axioms, and its correspondence to the physical world is entirely coincidental. Certainly, if Peano arithmetic didn’t correspond widely to real world problems, I would never have learned about it in high school and it might not even have been invented—but it remains true all the same.
This all just means that my idea of truth is different than yours—I think things can be true or false regardless of their predictive value. Specifically, I value statements of the form “If A, then (A worded slightly differently)” and think that almost all knowledge has that form. For example, “If the universe is consistent and objective, the scientific method will tend toward accurately describing the universe”. Once you introduce inductive reasoning, even for something as trivial as stating “the universe is consistent and objective”, then you introduce uncertainty—you switch from binary true/false to likely/unlikely and accurate/inaccurate and predictive/non-predictive.
If you equate “scientific type truth” with other people’s “actual truth” you will get into many pointless arguments. For example, you seem greatly offended that religious people sometimes disagree with science. For example, they will point out that science has always been wrong, and is probably still wrong. But they’ll probably agree that modern science more accurately and more precisely predicts about our world. In fact, you’ll probably get them to agree that science is perhaps the best method to make accurate and precise predictions. A statement doesn’t need to be true to be useful for making accurate predictions, for example Newtonian gravity. So why equate “makes accurate predictions” with “truth”?
No, the real world does not work via Peano arithmetic. Your experiments with apples and earplugs are simply applications of conservation of mass and immutability of inanimate objects, and other such principles. Before you learned such things, you were thrilled with the game of peek-a-boo—of how someone could cease to exist, and then appear out of nowhere.
Consider this experiment: Take 2 apples, cut them in half. Take 2 more apples, cut them in half. Put all together. How many apples do you have? The answer is not “4 apples”, the answer is “8 half-apples”. Furthermore, each individual apple remains the same apple as before (minus the effects of time), so that any differences in size, shape, coloration, bruising, etc would remain the same. Apples aren’t numbers, and can’t be substituted for each other.
The world abounds with examples where Peano arithmetic does not apply. Consider adding two speeds together—they do not add via Peano arithmetic, such that there exists a speed X, such that 2X + 2X = 3X. If we’re using naive multiplication, that speed is where X=c*sqrt(7/36). None of this changes my beliefs about Peano arithmetic—it is necessarily true given its axioms, and its correspondence to the physical world is entirely coincidental. Certainly, if Peano arithmetic didn’t correspond widely to real world problems, I would never have learned about it in high school and it might not even have been invented—but it remains true all the same.
This all just means that my idea of truth is different than yours—I think things can be true or false regardless of their predictive value. Specifically, I value statements of the form “If A, then (A worded slightly differently)” and think that almost all knowledge has that form. For example, “If the universe is consistent and objective, the scientific method will tend toward accurately describing the universe”. Once you introduce inductive reasoning, even for something as trivial as stating “the universe is consistent and objective”, then you introduce uncertainty—you switch from binary true/false to likely/unlikely and accurate/inaccurate and predictive/non-predictive.
If you equate “scientific type truth” with other people’s “actual truth” you will get into many pointless arguments. For example, you seem greatly offended that religious people sometimes disagree with science. For example, they will point out that science has always been wrong, and is probably still wrong. But they’ll probably agree that modern science more accurately and more precisely predicts about our world. In fact, you’ll probably get them to agree that science is perhaps the best method to make accurate and precise predictions. A statement doesn’t need to be true to be useful for making accurate predictions, for example Newtonian gravity. So why equate “makes accurate predictions” with “truth”?